<!DOCTYPE html>



  


<html class="theme-next pisces use-motion" lang="zh-Hans">
<head>
  <meta charset="UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1"/>
<meta name="theme-color" content="#222">


<meta name="google-site-verification" content="E9deYnivN5MuHMuIfiMZZfS0alv-d_0UjcwjBL79lGU" />



<meta name="baidu-site-verification" content="iHYWJxscwD" />










<meta http-equiv="Cache-Control" content="no-transform" />
<meta http-equiv="Cache-Control" content="no-siteapp" />



  <meta name="google-site-verification" content="true" />








  <meta name="baidu-site-verification" content="true" />







  
  
  <link href="/lib/fancybox/source/jquery.fancybox.css?v=2.1.5" rel="stylesheet" type="text/css" />







<link href="/lib/font-awesome/css/font-awesome.min.css?v=4.6.2" rel="stylesheet" type="text/css" />

<link href="/css/main.css?v=5.1.4" rel="stylesheet" type="text/css" />


  <link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png?v=5.1.4">


  <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png?v=5.1.4">


  <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png?v=5.1.4">


  <link rel="mask-icon" href="/images/logo.svg?v=5.1.4" color="#222">





  <meta name="keywords" content="Python,量化投资,机器学习,回归分析,一元线性回归," />










<meta name="description" content="变量之间的非确定性相关关系。一般形式:y &#x3D; f(x0,x1,x2,…xp)+ε若为线性回归，y &#x3D; β0+β1x1+β2x2+…+βnxn+εβ0，β1等为回归系数，ε为随机误差。模型假设①零均值，ε均值为0②同方差，ε项方差为常数③无自相关性，ε项值之间无自相关性④正态分布，ε项呈正态分布。⑤x1，x2等解释变量之间是非随机变量，其观测值是常数。⑥解释变量之间不存在精确线性关系。⑦样本个数多于">
<meta property="og:type" content="article">
<meta property="og:title" content="量化投资学习笔记15——回归分析:一元线性回归">
<meta property="og:url" content="https://zwdnet.github.io/2020/03/07/%E9%87%8F%E5%8C%96%E6%8A%95%E8%B5%84%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B015%E2%80%94%E2%80%94%E5%9B%9E%E5%BD%92%E5%88%86%E6%9E%90-%E4%B8%80%E5%85%83%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/index.html">
<meta property="og:site_name" content="赵瑜敏的口腔医学专业学习博客">
<meta property="og:description" content="变量之间的非确定性相关关系。一般形式:y &#x3D; f(x0,x1,x2,…xp)+ε若为线性回归，y &#x3D; β0+β1x1+β2x2+…+βnxn+εβ0，β1等为回归系数，ε为随机误差。模型假设①零均值，ε均值为0②同方差，ε项方差为常数③无自相关性，ε项值之间无自相关性④正态分布，ε项呈正态分布。⑤x1，x2等解释变量之间是非随机变量，其观测值是常数。⑥解释变量之间不存在精确线性关系。⑦样本个数多于">
<meta property="og:locale">
<meta property="og:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/01.png">
<meta property="og:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/02.png">
<meta property="og:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/03.png">
<meta property="og:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/04.png">
<meta property="og:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/wx.jpg">
<meta property="article:published_time" content="2020-03-07T05:45:16.000Z">
<meta property="article:modified_time" content="2020-08-30T05:52:04.000Z">
<meta property="article:author" content="赵瑜敏">
<meta property="article:tag" content="Python">
<meta property="article:tag" content="量化投资">
<meta property="article:tag" content="机器学习">
<meta property="article:tag" content="回归分析">
<meta property="article:tag" content="一元线性回归">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/01.png">



<script type="text/javascript" id="hexo.configurations">
  var NexT = window.NexT || {};
  var CONFIG = {
    root: '',
    scheme: 'Pisces',
    version: '5.1.4',
    sidebar: {"position":"left","display":"post","offset":12,"b2t":false,"scrollpercent":false,"onmobile":false},
    fancybox: true,
    tabs: true,
    motion: {"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},
    duoshuo: {
      userId: '0',
      author: '博主'
    },
    algolia: {
      applicationID: '',
      apiKey: '',
      indexName: '',
      hits: {"per_page":10},
      labels: {"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}
    }
  };
</script>



  <link rel="canonical" href="https://zwdnet.github.io/2020/03/07/量化投资学习笔记15——回归分析-一元线性回归/"/>





  <title>量化投资学习笔记15——回归分析:一元线性回归 | 赵瑜敏的口腔医学专业学习博客</title>
  








<meta name="generator" content="Hexo 5.4.0"></head>

<body itemscope itemtype="http://schema.org/WebPage" lang="zh-Hans">

  
  
    
  

  <div class="container sidebar-position-left page-post-detail">
    <div class="headband"></div>

    <header id="header" class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-wrapper">
  <div class="site-meta ">
    

    <div class="custom-logo-site-title">
      <a href="/"  class="brand" rel="start">
        <span class="logo-line-before"><i></i></span>
        <span class="site-title">赵瑜敏的口腔医学专业学习博客</span>
        <span class="logo-line-after"><i></i></span>
      </a>
    </div>
      
        <p class="site-subtitle"></p>
      
  </div>

  <div class="site-nav-toggle">
    <button>
      <span class="btn-bar"></span>
      <span class="btn-bar"></span>
      <span class="btn-bar"></span>
    </button>
  </div>
</div>

<nav class="site-nav">
  

  
    <ul id="menu" class="menu">
      
        
        <li class="menu-item menu-item-home">
          <a href="/%20" rel="section">
            
              <i class="menu-item-icon fa fa-fw fa-home"></i> <br />
            
            首页
          </a>
        </li>
      
        
        <li class="menu-item menu-item-tags">
          <a href="/tags/%20" rel="section">
            
              <i class="menu-item-icon fa fa-fw fa-tags"></i> <br />
            
            标签
          </a>
        </li>
      
        
        <li class="menu-item menu-item-categories">
          <a href="/categories/%20" rel="section">
            
              <i class="menu-item-icon fa fa-fw fa-th"></i> <br />
            
            分类
          </a>
        </li>
      
        
        <li class="menu-item menu-item-archives">
          <a href="/archives/%20" rel="section">
            
              <i class="menu-item-icon fa fa-fw fa-archive"></i> <br />
            
            归档
          </a>
        </li>
      

      
    </ul>
  

  
</nav>



 </div>
    </header>

    <main id="main" class="main">
      <div class="main-inner">
        <div class="content-wrap">
          <div id="content" class="content">
            

  <div id="posts" class="posts-expand">
    

  

  
  
  

  <article class="post post-type-normal" itemscope itemtype="http://schema.org/Article">
  
  
  
  <div class="post-block">
    <link itemprop="mainEntityOfPage" href="https://zwdnet.github.io/2020/03/07/%E9%87%8F%E5%8C%96%E6%8A%95%E8%B5%84%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B015%E2%80%94%E2%80%94%E5%9B%9E%E5%BD%92%E5%88%86%E6%9E%90-%E4%B8%80%E5%85%83%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="name" content="">
      <meta itemprop="description" content="">
      <meta itemprop="image" content="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/tx.jpg">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="赵瑜敏的口腔医学专业学习博客">
    </span>

    
      <header class="post-header">

        
        
          <h1 class="post-title" itemprop="name headline">量化投资学习笔记15——回归分析:一元线性回归</h1>
        

        <div class="post-meta">
          <span class="post-time">
            
              <span class="post-meta-item-icon">
                <i class="fa fa-calendar-o"></i>
              </span>
              
                <span class="post-meta-item-text">发表于</span>
              
              <time title="创建于" itemprop="dateCreated datePublished" datetime="2020-03-07T05:45:16+00:00">
                2020-03-07
              </time>
            

            

            
          </span>

          
            <span class="post-category" >
            
              <span class="post-meta-divider">|</span>
            
              <span class="post-meta-item-icon">
                <i class="fa fa-folder-o"></i>
              </span>
              
                <span class="post-meta-item-text">分类于</span>
              
              
                <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
                  <a href="/categories/%E9%87%8F%E5%8C%96%E6%8A%95%E8%B5%84/" itemprop="url" rel="index">
                    <span itemprop="name">量化投资</span>
                  </a>
                </span>

                
                
              
            </span>
          

          
            
          

          
          

          

          
            <div class="post-wordcount">
              
                
                  <span class="post-meta-divider">|</span>
                
                <span class="post-meta-item-icon">
                  <i class="fa fa-file-word-o"></i>
                </span>
                
                  <span class="post-meta-item-text">字数统计&#58;</span>
                
                <span title="字数统计">
                  2.1k
                </span>
              

              
                <span class="post-meta-divider">|</span>
              

              
                <span class="post-meta-item-icon">
                  <i class="fa fa-clock-o"></i>
                </span>
                
                  <span class="post-meta-item-text">阅读时长 &asymp;</span>
                
                <span title="阅读时长">
                  8
                </span>
              
            </div>
          

          

        </div>
      </header>
    

    
    
    
    <div class="post-body" itemprop="articleBody">

      
      

      
        <p>变量之间的非确定性相关关系。<br>一般形式:y = f(x0,x1,x2,…xp)+ε<br>若为线性回归，y = β0+β1x1+β2x2+…+βnxn+ε<br>β0，β1等为回归系数，ε为随机误差。<br>模型假设<br>①零均值，ε均值为0<br>②同方差，ε项方差为常数<br>③无自相关性，ε项值之间无自相关性<br>④正态分布，ε项呈正态分布。<br>⑤x1，x2等解释变量之间是非随机变量，其观测值是常数。<br>⑥解释变量之间不存在精确线性关系。<br>⑦样本个数多于解释变量个数。<br>建立回归分析模型的一般步骤<br>①需求分析明确变量<br>②数据收集加工，检查是否满足回归分析断的假设。<br>③确定回归模型:一元，二元，多元;线性，非线性。画散点图。<br>④确定模型参数:常用最小二乘法，若不符合假设条件，可用其它方法。<br>最小二乘法也叫最小化平方法，通过使误差的平方和最小来寻找最佳函数匹配。<br>⑤模型检验优化<br>⑥模型部署应用。<br>优点:模型简单，应用方便;有坚实的理论支撑;定量分析各变量间关系;模型预测结果可通过误差分析精确了解。<br>缺点:假设条件比较多且严格;变量选择对模型影响较大。<br>回归模型的参数估计<br>一元线性回归模型:研究与现象关系最大的主要因素，两者有密切关系，但并非确定性关系，可用该模型。<br>y = β0+β1x+ε<br>y为被解释变量或因变量，x为解释变量或自变量。β0回归常数，β1回归系数，二者统称为回归参数。ε为随机误差。其中E(ε) = 0，var(ε) = 常数。<br>一元线性回归方程:y = β0+β1x 忽略随机误差。<br>回归方程从平均意义上表达了变量y与x的统计规律性。<br>回归分析的主要任务是通过n组样本的观察值，对β0，β1进行估计，得到最终方程。<br>参数估计:最小二乘估计(OLE)<br>通过最小二乘法求出对β0，β1的估计值。<br>离差平方和Q(β0,β1)=Σ(yi - E(yi))²=Σ(yi -β0+β1xi)²<br>估计值满足使上式离差平方和最小。<br>其最小值的求法为求其偏导数，并令其为0。求解方程组即可。<br>实操一下。<br>先创建数据，画散点图。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">x = [<span class="number">3.4</span>, <span class="number">1.8</span>, <span class="number">4.6</span>, <span class="number">2.3</span>, <span class="number">3.1</span>, <span class="number">5.5</span>, <span class="number">0.7</span>, <span class="number">3</span>, <span class="number">2.6</span>, <span class="number">4.3</span>, <span class="number">2.1</span>, <span class="number">1.1</span>, <span class="number">6.1</span>, <span class="number">4.8</span>, <span class="number">3.8</span>]</span><br><span class="line">y = [<span class="number">26.2</span>, <span class="number">17.8</span>, <span class="number">31.3</span>, <span class="number">23.1</span>, <span class="number">27.5</span>, <span class="number">36</span>, <span class="number">14.1</span>, <span class="number">22.3</span>, <span class="number">19.6</span>, <span class="number">31.3</span>, <span class="number">24</span>, <span class="number">17.3</span>, <span class="number">43.2</span>, <span class="number">36.4</span>, <span class="number">26.1</span>]</span><br><span class="line">x = np.array(x)</span><br><span class="line">y = np.array(y)</span><br><span class="line">print(<span class="built_in">len</span>(x), <span class="built_in">len</span>(y))</span><br><span class="line">plt.scatter(x, y)</span><br><span class="line">plt.savefig(<span class="string">&quot;scatter.png&quot;</span>)</span><br></pre></td></tr></table></figure>
<p><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/01.png"><br>建立线性回归模型</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">df pd.DataFrame()</span><br><span class="line">df[<span class="string">&quot;X&quot;</span>] = x</span><br><span class="line">df[<span class="string">&quot;Y&quot;</span>] = y</span><br><span class="line">print(df)</span><br><span class="line">regr = linear_model.LinearRegression()</span><br></pre></td></tr></table></figure>
<p> 拟合</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">regr.fit(x.reshape(-<span class="number">1</span>,<span class="number">1</span>), y)</span><br></pre></td></tr></table></figure>
<p> 得到回归参数的二乘法估计</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">a, b = regr.coef_, regr.intercept_</span><br><span class="line">print(a, b)</span><br></pre></td></tr></table></figure>
<p> 画出拟合的直线</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">yp = a*x + b</span><br><span class="line">plt.scatter(x, y)</span><br><span class="line">plt.plot(x, yp)</span><br><span class="line">plt.savefig(<span class="string">&quot;fit.png&quot;</span>)</span><br></pre></td></tr></table></figure>
<p>拟合结果<br>a = 4.91933073 b = 10.277928549524688<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/02.png"><br>另一种参数估计方法:最大似然估计(MLE)<br>利用总体的分布密度或概率分布的表达式及其样本所提供的信息求未知参数估计量的一种方法。<br>基本思路:已知样本符合某种分布，但分布参数未知，通过实验来估计分布参数。估算思路:如果某组参数能使当前样本出现的概率最大，就认为该参数为最终的估计值。解决的是模型已定，参数未知的问题，用已知样本去估算未知参数。<br>出现当前情形的概率为f(x1x2x3…xn|θ)=f(x1|θ)f(x2|θ)f(x3|θ)…f(xn|θ)，其中θ未知。<br>似然函数L(θ|x1x2x3…xn) = f(x1x2x3…xn|θ)=f(x1|θ)f(x2|θ)f(x3|θ)…f(xn|θ)=Πf(xi|θ)<br>前提是事件x之间是独立的。<br>取对数lnL(θ|x1x2x3…xn) = lnf(x1|θ)+lnf(x2|θ)+lnf(x3|θ)+…+lnf(xn|θ) = Σf(xi|θ)<br>平均对数似然l = lnL(θ|x1x2x3…xn)/n<br>最大似然估计就是找到一个θ使得l最大。求导求出最大值。<br>一元线性回归，OLE和MLE是等价的，MLE还可以估计方差σ²的值。<br>无偏估计，估计不同样本，偏差平均值为0。反之则为有偏估计。<br>一元线性回归方差的性质:<br>①线性:估计回归参数为随机变量yi的线性函数。<br>②无偏，估计值y^为真实值的无偏估计，即E(y^) = E(y)<br>③参数的方差，与样本方差，随机误差的方差等有关系。<br>模型的检验<br>①回归系数是否显著:t检验<br>检验因变量y与自变量x之间是否真的存在线性关系?即β1=0?用t检验进行判断。<br>确定假设:目的是找到不达标的证据。原假设H0:β0=0，备择假设H1:β0≠0<br>检验水平:α=0.05或0.01<br>构造统计量:H0成立时，β1满足正态分布。<br>计算p值。<br>得出结论。<br>②回归方程是否显著:F检验。<br>根据平方和分解式，直接从回归效果检验回归方程的显著性。<br>③相关系数显著性检验:t检验。<br>样本相关系数可以作为总体相关系数的估计值，样本相关系数不等于零时需要进行假设检验确定是来自抽样误差还是来自整体。<br>④决定系数<br>SST = SSR + SSE, SSR占的比重越大，线性回归效果越好。<br>r² = SSR/SST<br>用模型的score函数来给模型评分，结果为0.9234781689805285。<br>还可以用statsmodels库来做，回归结果类似，与sklearn库不同的是它可以输出检验结果。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 用statsmodels来做</span></span><br><span class="line"> <span class="keyword">import</span> statsmodels.api <span class="keyword">as</span> sm</span><br><span class="line"> X = sm.add_constant(x)</span><br><span class="line"> model = sm.OLS(y, X)</span><br><span class="line"> result = model.fit()</span><br><span class="line"> print(<span class="string">&quot;statsmodels做线性回归&quot;</span>)</span><br><span class="line"> print(result.params)</span><br><span class="line"> print(result.summary())</span><br></pre></td></tr></table></figure>
<p>结果及解释(以下参考:<a target="_blank" rel="noopener" href="https://blog.csdn.net/CoderPai/article/details/83899268">https://blog.csdn.net/CoderPai/article/details/83899268</a>)<br>OLS Regression Results =====================================<br> Dep. Variable: y<br> R-squared: 0.923<br> Model: OLS<br> Adj. R-squared: 0.918<br> Method: Least Squares<br> F-statistic: 156.9<br> Date: Fri, 14 Feb 2020<br> Prob (F-statistic): 1.25e-08<br> Time: 15:48:01<br> Log-Likelihood: -32.811<br> No. Observations: 15<br> AIC: 69.62<br> Df Residuals: 13<br> BIC: 71.04<br> Df Model: 1<br>Covariance Type: nonrobust<br>coef  std err    t    P&gt;|t|      [0.025 0.975]              </p>
<p>const  10.2779    1.420      7.237      0.000       7.210      13.346<br>x1        4.9193      0.393     12.525      0.000       4.071       5.768           </p>
<p>Omnibus:                              2.551<br>Durbin-Watson:                   1.318              Prob(Omnibus):                  0.279   </p>
<p>Jarque-Bera (JB):                1.047              Skew:                          -0.003   Prob(JB):                        0.592              Kurtosis:                       1.706   Cond. No.                                  9.13<br> Adj. R-squared: 0.918 R平方，反映模型的拟合度，取值0-1，越高越好。<br>const coef:10.2779 y截距，即b值。<br>x1:4.9193 a值，即一元线性方程的一次项。<br>std err，反映系数的准确度，越低，准确度越高。<br>P&gt;|t|:p值，判断回归参数是否有统计学意义。本处均为0。<br>Confidence Interval：这是置信区间，表示我们的系数可能下降的范围（可能性为 95%）<br>另外回归方程的F值F-statistic:156.9，概率Prob (F-statistic):1.25e-08，小于0.05，回归分析效果是显著的。<br>残差(residual):回归值与真实值的差额。<br>ei = yi^-yi<br>残差是误差的估计值。<br>残差图，以x为横轴，残差为纵轴做的图。残差应在0附近随机变动，且变动不大。<br>残差图的几种情况。<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/03.png"><br>算出上面回归的残差并绘出残差图。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">residual = []</span><br><span class="line"><span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(<span class="built_in">len</span>(x)):</span><br><span class="line"> cha = res[<span class="number">0</span>] + res[<span class="number">1</span>]*x[i] - y[i]</span><br><span class="line"> residual.append(cha)</span><br><span class="line">print(residual)</span><br><span class="line">plt.scatter(x, residual)</span><br><span class="line">plt.savefig(<span class="string">&quot;residual.png&quot;</span>)</span><br></pre></td></tr></table></figure>
<p><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0178-QTLearn/10/04.png"><br>可以看到残差图还是比较正常的。<br>模型的应用，有预测和控制。<br>预测分为单值预测和区间预测。<br>单值预测直接求值。<br>区间预测:以α显著水平，找到区间(T1,T2)，使得某特定X0的实际值Y0以1-α的概率在该区间内。即p(T1&lt;Y0&lt;T2) = 1-α。<br>控制:要把因变量控制在一定范围内，求自变量的取值范围。<br>下次学习多元回归。</p>
<p>我发文章的四个地方，欢迎大家在朋友圈等地方分享，欢迎点“在看”。<br>我的个人博客地址：<a href="https://zwdnet.github.io/">https://zwdnet.github.io</a><br>我的知乎文章地址： <a target="_blank" rel="noopener" href="https://www.zhihu.com/people/zhao-you-min/posts">https://www.zhihu.com/people/zhao-you-min/posts</a><br>我的博客园博客地址： <a target="_blank" rel="noopener" href="https://www.cnblogs.com/zwdnet/">https://www.cnblogs.com/zwdnet/</a><br>我的微信个人订阅号：赵瑜敏的口腔医学学习园地</p>
<p><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/wx.jpg"></p>

      
    </div>
    
    
    

    

    
      <div>
        <div style="padding: 10px 0; margin: 20px auto; width: 90%; text-align: center;">
  <div>欢迎打赏！感谢支持！</div>
  <button id="rewardButton" disable="enable" onclick="var qr = document.getElementById('QR'); if (qr.style.display === 'none') {qr.style.display='block';} else {qr.style.display='none'}">
    <span>打赏</span>
  </button>
  <div id="QR" style="display: none;">

    
      <div id="wechat" style="display: inline-block">
        <img id="wechat_qr" src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/mm_facetoface_collect_qrcode_1542944836634.png" alt=" 微信支付"/>
        <p>微信支付</p>
      </div>
    

    
      <div id="alipay" style="display: inline-block">
        <img id="alipay_qr" src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/1542944857770.jpg" alt=" 支付宝"/>
        <p>支付宝</p>
      </div>
    

    

  </div>
</div>

      </div>
    

    

    <footer class="post-footer">
      
        <div class="post-tags">
          
            <a href="/tags/Python/" rel="tag"># Python</a>
          
            <a href="/tags/%E9%87%8F%E5%8C%96%E6%8A%95%E8%B5%84/" rel="tag"># 量化投资</a>
          
            <a href="/tags/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/" rel="tag"># 机器学习</a>
          
            <a href="/tags/%E5%9B%9E%E5%BD%92%E5%88%86%E6%9E%90/" rel="tag"># 回归分析</a>
          
            <a href="/tags/%E4%B8%80%E5%85%83%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/" rel="tag"># 一元线性回归</a>
          
        </div>
      

      
      
      

      
        <div class="post-nav">
          <div class="post-nav-next post-nav-item">
            
              <a href="/2020/03/07/%E6%A0%B9%E5%B0%96%E7%89%99%E7%89%87%E6%8B%8D%E6%91%84%E6%8A%80%E5%B7%A7/" rel="next" title="根尖牙片拍摄技巧">
                <i class="fa fa-chevron-left"></i> 根尖牙片拍摄技巧
              </a>
            
          </div>

          <span class="post-nav-divider"></span>

          <div class="post-nav-prev post-nav-item">
            
              <a href="/2020/03/07/%E6%A5%94%E7%BC%BA%E7%9A%84%E8%84%B1%E6%95%8F%E3%80%81%E4%BF%AE%E5%A4%8D%E6%B2%BB%E7%96%97/" rel="prev" title="楔缺的脱敏、修复治疗">
                楔缺的脱敏、修复治疗 <i class="fa fa-chevron-right"></i>
              </a>
            
          </div>
        </div>
      

      
      
    </footer>
  </div>
  
  
  
  </article>



    <div class="post-spread">
      
    </div>
  </div>


          </div>
          


          

  
    <div class="comments" id="comments">
      <div id="lv-container" data-id="city" data-uid="MTAyMC80MTA2Mi8xNzU4Nw=="></div>
    </div>

  



        </div>
        
          
  
  <div class="sidebar-toggle">
    <div class="sidebar-toggle-line-wrap">
      <span class="sidebar-toggle-line sidebar-toggle-line-first"></span>
      <span class="sidebar-toggle-line sidebar-toggle-line-middle"></span>
      <span class="sidebar-toggle-line sidebar-toggle-line-last"></span>
    </div>
  </div>

  <aside id="sidebar" class="sidebar">
    
    <div class="sidebar-inner">

      

      

      <section class="site-overview-wrap sidebar-panel sidebar-panel-active">
        <div class="site-overview">
          <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
            
              <img class="site-author-image" itemprop="image"
                src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/other/tx.jpg"
                alt="" />
            
              <p class="site-author-name" itemprop="name"></p>
              <p class="site-description motion-element" itemprop="description"></p>
          </div>

          <nav class="site-state motion-element">

            
              <div class="site-state-item site-state-posts">
              
                <a href="/archives/%20%7C%7C%20archive">
              
                  <span class="site-state-item-count">452</span>
                  <span class="site-state-item-name">日志</span>
                </a>
              </div>
            

            
              
              
              <div class="site-state-item site-state-categories">
                <a href="/categories/index.html">
                  <span class="site-state-item-count">29</span>
                  <span class="site-state-item-name">分类</span>
                </a>
              </div>
            

            
              
              
              <div class="site-state-item site-state-tags">
                <a href="/tags/index.html">
                  <span class="site-state-item-count">544</span>
                  <span class="site-state-item-name">标签</span>
                </a>
              </div>
            

          </nav>

          

          

          
          

          
          

          

        </div>
      </section>

      

      

    </div>
  </aside>


        
      </div>
    </main>

    <footer id="footer" class="footer">
      <div class="footer-inner">
        <div class="copyright">&copy; <span itemprop="copyrightYear">2021</span>
  <span class="with-love">
    <i class="fa fa-user"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">本站版权归赵瑜敏所有，如欲转载请与本人联系。</span>

  
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item-icon">
      <i class="fa fa-area-chart"></i>
    </span>
    
      <span class="post-meta-item-text">Site words total count&#58;</span>
    
    <span title="Site words total count">1225.8k</span>
  
</div>









<div>
  <script type="text/javascript">var cnzz_protocol = (("https:" == document.location.protocol) ? " https://" : " http://");document.write(unescape("%3Cspan id='cnzz_stat_icon_1275447216'%3E%3C/span%3E%3Cscript src='" + cnzz_protocol + "s11.cnzz.com/z_stat.php%3Fid%3D1275447216%26online%3D1%26show%3Dline' type='text/javascript'%3E%3C/script%3E"));</script>
</div>

        







  <div style="display: none;">
    <script src="//s95.cnzz.com/z_stat.php?id=1275447216&web_id=1275447216" language="JavaScript"></script>
  </div>



        
      </div>
    </footer>

    
      <div class="back-to-top">
        <i class="fa fa-arrow-up"></i>
        
      </div>
    

    

  </div>

  

<script type="text/javascript">
  if (Object.prototype.toString.call(window.Promise) !== '[object Function]') {
    window.Promise = null;
  }
</script>









  












  
  
    <script type="text/javascript" src="/lib/jquery/index.js?v=2.1.3"></script>
  

  
  
    <script type="text/javascript" src="/lib/fastclick/lib/fastclick.min.js?v=1.0.6"></script>
  

  
  
    <script type="text/javascript" src="/lib/jquery_lazyload/jquery.lazyload.js?v=1.9.7"></script>
  

  
  
    <script type="text/javascript" src="/lib/velocity/velocity.min.js?v=1.2.1"></script>
  

  
  
    <script type="text/javascript" src="/lib/velocity/velocity.ui.min.js?v=1.2.1"></script>
  

  
  
    <script type="text/javascript" src="/lib/fancybox/source/jquery.fancybox.pack.js?v=2.1.5"></script>
  


  


  <script type="text/javascript" src="/js/src/utils.js?v=5.1.4"></script>

  <script type="text/javascript" src="/js/src/motion.js?v=5.1.4"></script>



  
  


  <script type="text/javascript" src="/js/src/affix.js?v=5.1.4"></script>

  <script type="text/javascript" src="/js/src/schemes/pisces.js?v=5.1.4"></script>



  
  <script type="text/javascript" src="/js/src/scrollspy.js?v=5.1.4"></script>
<script type="text/javascript" src="/js/src/post-details.js?v=5.1.4"></script>



  


  <script type="text/javascript" src="/js/src/bootstrap.js?v=5.1.4"></script>



  


  




	





  





  
    <script type="text/javascript">
      (function(d, s) {
        var j, e = d.getElementsByTagName(s)[0];
        if (typeof LivereTower === 'function') { return; }
        j = d.createElement(s);
        j.src = 'https://cdn-city.livere.com/js/embed.dist.js';
        j.async = true;
        e.parentNode.insertBefore(j, e);
      })(document, 'script');
    </script>
  












  





  

  

  

  
  

  

  

  

  
</body>
</html>
